Explicit bounds are given on the norms of prime ideals generating arbitrary subgroups of
ray class groups of number fields, assuming the extended Riemann hypothesis. These are
the first explicit bounds for this problem and are significantly better than previously
known asymptotic bounds. Applied to the integers, they express that any subgroup of
index
of the multiplicative group of integers modulo
is generated by prime
numbers smaller than
,
subject to the Riemann hypothesis. Two particular consequences relate to
mathematical cryptology. Applied to cyclotomic fields, they provide explicit
bounds on generators of the relative class group, needed in some previous
work on the shortest vector problem on ideal lattices. Applied to Jacobians
of hyperelliptic curves, they allow one to derive bounds on the degrees of
isogenies required to make their horizontal isogeny graphs connected. Such
isogeny graphs are used to study the discrete logarithm problem on said
Jacobians.
Keywords
class group, ray class group, prime ideal, isogeny graph
